Nature | Vol 581 | 14 May 2020 | 155their uncertainties are essential. The ab initio transition frequency
fi(theor) of each hyperfine component i is the sum of two contributions,
ffspinavg+ i
(theor)
spin,
(theor). The dominant contribution isf(tspheinorav)g=1,314,925,752.896( 18 )(theory 61 )kCODATA 2018 Hz (1)computed^15 including all relativistic and radiative corrections up to
the relative order α^5 and partially including contributions of the order
α^6 (Table 2 ). The value fsp(theinorav)g is updated from the value reported in
ref.^16 by using CODATA 2018^29 updates of the Rydberg constant, the
particle masses (in atomic mass units, u), the proton charge radius
and the deuteron charge radius. The theory uncertainty is estimated
as u(fsp(theinorav)g) ≈ 0.018 kHz, while the larger CODATA 2018 uncertainty,
uCODATA2018(fsp(theinorav)g) ≈ 0.061 kHz, is dominated by the uncertainties of
the particle masses.
A spin frequency contribution fsp(theinor,i) is the difference of the spin
structure energies of the upper and lower spin states involved in the
transition. For the favoured transitions measured here, the values of
fsp(thein,ori) are of the order of 10 MHz. The spin contributions are computed
by diagonalizing the Breit–Pauli spin Hamiltonian of ref.^35. The various
terms of this Hamiltonian are proportional to coefficients Ekk,′E, com-
puted ab initio (Extended Data Table 1). The spin Hamiltonian of the
N = 0 level necessitates two coefficients, E 4 and E 5 , while the N = 1 level
necessitates nine, E′, ..., 19 E′.
The coefficients E 4 , E′ 4 and E 5 , E′ 5 describe the dominant se⋅Ip and se⋅Id
interactions, respectively, and have been calculated with high theo-
retical precision, including all corrections of the order α^2 EF/h and the
leading corrections of the order α^3 EF/h, where EF ≈ h(1.4 GHz) is the
Fermi contact energy for the hyperfine splitting in atomic hydrogen
and h is Planck’s constant^38. The fractional theoretical uncertainties of
these spin Hamiltonian coefficients are of the order α^3 ; they are estimated
as εF = 1 × 10−6. Furthermore, the signed theory errors are expected to be
nearly equal: Δ≈EE 4 (theor) Δ′ 4 (theor) and Δ≈EE 5 (theor) Δ′ 5 (theor) (Methods).
The other spin coefficients, E′ 1 , E′ 2 , E′ 3 , E′ 6 , E′ 7 , E′ 8 and E′ 9 , have been
obtained within the Breit–Pauli approximation. We computed them
using our most precise non-relativistic non-adiabatic molecular vari-
ational wave functions (Methods, Extended Data Table 1). The omitted
terms are of the relative order α^2. References^38 ,^39 lead us to estimate a
common fractional theory uncertainty equal to α^2 = ε 0 ≈ 5 × 10−5.
To determine the impact of the theory uncertainty of a particular
Hamiltonian coefficient on a particular spin frequency, we introduce the
quantities γik′Δ, E′k(theor), with the derivatives γE′=ik, ∂′spin,1ik(′EE,...,′ 9 )/∂′E
relevant for the upper spin level and similarly for the lower spin level. The
γ values are reported in Extended Data Table 1. Assuming equal theory
errors for the pairs (E 4 , E′ 4 ) and (E 5 , E′ 5 ), we conservatively estimate the total
theory uncertainty of the spin-frequency contribution with the following
expressionuf()(tspheinor,ii) =|εγF∑∑′′kEEk−|γεikk+|γ′′ikEk|
4, 5,,^0
1,2,3,6,7,8, 9,The form of the first sum embodies the assumption of equal fractional
errors and correlation, Δ=EE(t4,he 5 or) δε4,5F4, 5 , Δ′EE4,(t 5 heor)=′δε4,5F4, 5 , with
δ 45 =1or−1,=δ 1or− 1. The similarities γγ 44 ≈′ and γγ 55 ≈′ for the lower
and upper rotational levels then lead to a strong suppression of the con-
tributions related to the theory errors of E 4 , E′ 4 , E 5 and E′ 5. This results in
the spin-frequency uncertainties shown in Table 1 (column 6). They
dominate the total uncertainty of the transition frequencies fi(theor).Comparison between theory and experiment
Table 1 presents the comparison between the theory and experimental
data of the individual hyperfine components of the rotationalTable 1 | Experimental rotational frequencies, and comparison with theoretical ab initio frequencies
Line i G 1 G 2 F → GG 12 ′′F′ fi(exp) u()f()iexp fi()theor u()fsp()thineo,ir u()fsp()thineoavrg uCODATA()fi()theor
12 122 → 121 1314892544.276 0.040 1314892544.23 1.2 0.018 0.061
14 100 → 101 1314916678.487 0.064 1314916678.74 1.3 0.018 0.061
16 011 → 012 1314923618.028 0.017 1314923617.94 0.20 0.018 0.061
19 122 → 123 1314935827.695 0.037 1314935827.58 1.2 0.018 0.061
20 122 → 122 1314937488.614 0.060 1314937488.80 1.4 0.018 0.061
21 111 → 112 1314937540.762 0.046 1314937540.61 0.73 0.018 0.061
Uncertainties are denoted by u. Frequency values are in kHz. The theoretical values fi(theor) were computed using CODATA 2018 constants. The last three columns show the three contributions to
the total uncertainty of fi(theor). Line 16 offers the most stringent comparison, due to its comparatively small theory uncertainty.
Table 2 | Contributions to the ab initio spin-averaged rotational frequency fspin()theoavrg
Term Relative
order
Contribution (kHz) Originf(0) 1 1,314,886,776.526 Solution of three-body Schrödinger equation
f(2) α^2 48,416.268 Relativistic corrections in Breit–Pauli approximation; nuclear radii
f(3) α^3 −9,378.119Leading-order radiative corrections (for example, leading-order Lamb shift, anomalous magnetic moment)
f(4) α^4 −65.631(2) One-loop, two-loop radiative corrections; relativistic corrections
f(5) α^5 3.923(3) Radiative corrections up to three-loop diagrams; Wichman–Kroll contribution
f(6) α^6 −0.070(18) Higher-order radiative corrections
Total (^) fsp()thineoavrg 1,314,925,752.896(18)
The values were calculated using CODATA 2018 values of the fundamental constants. The main contribution f(0) is of order cR∞(me/μpd). Recoil corrections (due to finite masses of nuclei) are
included fully at the order α^2 ; the leading recoil corrections proportional to me/mp or me/md are included at the order α^3. Contributions due to the finite size of the nuclei are included in the
f(2) term^15. The one-loop contribution from μ+–μ− vacuum polarization is included in f(3). The estimated fractional theory uncertainty of the spin-averaged frequency is ur = 1.4 × 10−11
(u f(sp(theinor)-avg) = 0.018 kHz). The impact of the fundamental constants’ uncertainties is given in the text. The change in the value of f(0) from CODATA 2014 to CODATA 2018 has contributions of
−0.041 kHz from the Rydberg constant adjustment and 0.213 kHz from the particle masses adjustments. The change in the value of f(2) due to the proton and deuteron charge radii
adjustments is 0.104 kHz.